Nonparametric comparison of quantile curves: a stochastic process approach

Abstract

A new test for comparing conditional quantile curves is proposed which is able to detect Pitman alternatives converging to the null hypothesis at the optimal rate. The basic idea of the test is to measure differences between the curves by a process of integrated nonparametric estimates of the quantile curve. We prove weak convergence of this process to a Gaussian process and study the finite sample properties of a Kolmogorov–Smirnov test by means of a simulation study.

AMS Subject Classification::

• 62G10

Keywords:

• nonparametric analysis of covariance
• quantile regression
• crossing quantile curves
• monotone rearrangements

Acknowledgements

This work has been supported in part by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823, Teilprojekt C1) of the German Research Foundation (DFG). We thank two unknown referees for their constructive comments on an earlier version of this paper, Betina Berghaus for running some of the simulations, and Martina Stein, who typed parts of this paper with considerable technical expertise.